Properties

Label 10.18.a.c
Level $10$
Weight $18$
Character orbit 10.a
Self dual yes
Analytic conductor $18.322$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,18,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.3222087345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{83281}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 20820 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 30\sqrt{83281}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 256 q^{2} + ( - \beta - 654) q^{3} + 65536 q^{4} + 390625 q^{5} + (256 \beta + 167424) q^{6} + ( - 2907 \beta + 301922) q^{7} - 16777216 q^{8} + (1308 \beta - 53759547) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 256 q^{2} + ( - \beta - 654) q^{3} + 65536 q^{4} + 390625 q^{5} + (256 \beta + 167424) q^{6} + ( - 2907 \beta + 301922) q^{7} - 16777216 q^{8} + (1308 \beta - 53759547) q^{9} - 100000000 q^{10} + (5238 \beta - 235740648) q^{11} + ( - 65536 \beta - 42860544) q^{12} + (87228 \beta - 770917114) q^{13} + (744192 \beta - 77292032) q^{14} + ( - 390625 \beta - 255468750) q^{15} + 4294967296 q^{16} + ( - 4460868 \beta + 16069950282) q^{17} + ( - 334848 \beta + 13762444032) q^{18} + (7351668 \beta + 64336264700) q^{19} + 25600000000 q^{20} + (1599256 \beta + 217690623312) q^{21} + ( - 1340928 \beta + 60349605888) q^{22} + ( - 7606701 \beta + 325179929646) q^{23} + (16777216 \beta + 10972299264) q^{24} + 152587890625 q^{25} + ( - 22330368 \beta + 197354781184) q^{26} + (182044278 \beta + 21578017140) q^{27} + ( - 190513152 \beta + 19786760192) q^{28} + ( - 444821544 \beta + 1271527374990) q^{29} + (100000000 \beta + 65400000000) q^{30} + (461196486 \beta + 3919703999372) q^{31} - 1099511627776 q^{32} + (232314996 \beta - 238428906408) q^{33} + (1141982208 \beta - 4113907272192) q^{34} + ( - 1135546875 \beta + 117938281250) q^{35} + (85721088 \beta - 3523185672192) q^{36} + ( - 818505288 \beta - 13902604548778) q^{37} + ( - 1882027008 \beta - 16470083763200) q^{38} + (713870002 \beta - 6033811768644) q^{39} - 6553600000000 q^{40} + (4473032724 \beta - 18913182132078) q^{41} + ( - 409409536 \beta - 55728799567872) q^{42} + ( - 7033515453 \beta + 14815226963426) q^{43} + (343277568 \beta - 15449499107328) q^{44} + (510937500 \beta - 20999823046875) q^{45} + (1947315456 \beta - 83246061989376) q^{46} + (18068763213 \beta + 110395791511002) q^{47} + ( - 4294967296 \beta - 2808908611584) q^{48} + ( - 1755374508 \beta + 400861292338977) q^{49} - 39062500000000 q^{50} + ( - 13152542610 \beta + 323845245632772) q^{51} + (5716574208 \beta - 50522823983104) q^{52} + ( - 14912836812 \beta + 529153008800286) q^{53} + ( - 46603335168 \beta - 5523972387840) q^{54} + (2046093750 \beta - 92086190625000) q^{55} + (48771366912 \beta - 5065410609152) q^{56} + ( - 69144255572 \beta - 593104753551000) q^{57} + (113874315264 \beta - 325511007997440) q^{58} + (122690464632 \beta + 763910358617580) q^{59} + ( - 25600000000 \beta - 16742400000000) q^{60} + ( - 134737202448 \beta - 315869106561058) q^{61} + ( - 118066300416 \beta - 10\!\cdots\!32) q^{62}+ \cdots + ( - 589941274770 \beta + 13\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{2} - 1308 q^{3} + 131072 q^{4} + 781250 q^{5} + 334848 q^{6} + 603844 q^{7} - 33554432 q^{8} - 107519094 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 512 q^{2} - 1308 q^{3} + 131072 q^{4} + 781250 q^{5} + 334848 q^{6} + 603844 q^{7} - 33554432 q^{8} - 107519094 q^{9} - 200000000 q^{10} - 471481296 q^{11} - 85721088 q^{12} - 1541834228 q^{13} - 154584064 q^{14} - 510937500 q^{15} + 8589934592 q^{16} + 32139900564 q^{17} + 27524888064 q^{18} + 128672529400 q^{19} + 51200000000 q^{20} + 435381246624 q^{21} + 120699211776 q^{22} + 650359859292 q^{23} + 21944598528 q^{24} + 305175781250 q^{25} + 394709562368 q^{26} + 43156034280 q^{27} + 39573520384 q^{28} + 2543054749980 q^{29} + 130800000000 q^{30} + 7839407998744 q^{31} - 2199023255552 q^{32} - 476857812816 q^{33} - 8227814544384 q^{34} + 235876562500 q^{35} - 7046371344384 q^{36} - 27805209097556 q^{37} - 32940167526400 q^{38} - 12067623537288 q^{39} - 13107200000000 q^{40} - 37826364264156 q^{41} - 111457599135744 q^{42} + 29630453926852 q^{43} - 30898998214656 q^{44} - 41999646093750 q^{45} - 166492123978752 q^{46} + 220791583022004 q^{47} - 5617817223168 q^{48} + 801722584677954 q^{49} - 78125000000000 q^{50} + 647690491265544 q^{51} - 101045647966208 q^{52} + 10\!\cdots\!72 q^{53}+ \cdots + 26\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
144.792
−143.792
−256.000 −9311.53 65536.0 390625. 2.38375e6 −2.48655e7 −1.67772e7 −4.24355e7 −1.00000e8
1.2 −256.000 8003.53 65536.0 390625. −2.04890e6 2.54694e7 −1.67772e7 −6.50836e7 −1.00000e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.18.a.c 2
3.b odd 2 1 90.18.a.k 2
4.b odd 2 1 80.18.a.d 2
5.b even 2 1 50.18.a.f 2
5.c odd 4 2 50.18.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.c 2 1.a even 1 1 trivial
50.18.a.f 2 5.b even 2 1
50.18.b.d 4 5.c odd 4 2
80.18.a.d 2 4.b odd 2 1
90.18.a.k 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1308T_{3} - 74525184 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1308 T - 74525184 \) Copy content Toggle raw display
$5$ \( (T - 390625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 633309492538016 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 57\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 34\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 20\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 77\!\cdots\!16 \) Copy content Toggle raw display
show more
show less